#9706773

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#46749	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10,15,13)(11,17,12,16)(14,18,22,21)(19,24,20,23)(25,26,30,32)(27,28,35,31)(29,33,38,40)(34,36,37,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,12)(10,16,13,17)(14,19,22,20)(18,23,21,24)(25,27)(26,31)(28,32)(29,37)(30,35)(33,36)(34,38)(39,40);(1,3,5,4)(2,6,7,8)(26,32)(28,31)(33,40)(36,39);(9,12,15,11)(10,17,13,16)(14,20,22,19)(18,24,21,23)(25,28,30,31)(26,27,32,35)(29,36,38,39)(33,34,40,37);(1,4,5,3)(2,6,7,8)(9,13,15,10)(11,17,12,16)(14,18,22,21)(19,23,20,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37);(1,2)(3,6)(4,8)(5,7)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,27)(26,31)(28,32)(29,37)(30,35)(33,36)(34,38)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9613969);